PERFEZIONAMENTO IN MATEMATICA FINANZIARIA E ATTUARIALE. The impact of contagion on large portfolios. Modeling aspects.


 Cory Wells
 2 years ago
 Views:
Transcription
1 SCUOLA ORMALE SUPERIORE PISA PERFEZIOAMETO I MATEMATICA FIAZIARIA E ATTUARIALE TESI DI PERFEZIOAMETO The impact of contagion on large portfolios. Modeling aspects. Perfezionando: Marco Tolotti Relatori: Ch.mo Prof. Paolo Dai Pra Ch.mo Prof. Wolfgang J. Runggaldier
2
3 A Maria e Federico
4
5 Acknowledgments I must rst express my deepest gratitude towards Paolo Dai Pra and Wolfgang Runggaldier for their precious help and their courtesy in reading countless versions of this thesis. Their suggestions and guidance have always encouraged me throughout the study. A very special thanks goes out to Elena Sartori. I cannot forget the beautiful hours spent discussing about the generators of the Markov processes. I am very grateful for the valuable comments of Rama Cont and Rüdiger Frey. I wish to thank Maurizio Pratelli for his patience and kindness, the Amici della Scuola ormale and the Scuola ormale Superiore for the nancial support. I was delighted to interact with the Finance group at the Department of Mathematics of the ETH Zurich and the Swiss Banking Institute of the University of Zurich during the Master of Advanced Studies in Finance. I would like to thank in particular Philipp Schönbucher for giving me the opportunity to work with him for the Master thesis. My gratitude goes also to Fulvio Ortu for his trust in oering me a position at the Bocconi University. I must acknowledge the (former) Istituto di Metodi Quantitativi and the Department of Finance at the Bocconi university for the nancial support. Anna Battauz, Elena Catanese, Francesca Beccacece, Giuliana Bordigoni, Marzia De Donno, Claudio Tebaldi, Fabio Maccheroni, Francesco Corielli, Gianluca Fusai and Gino Favero have been supporting me with inspiring discussions and their friendship.... no words to express how I am grateful to Maria and my parents. Milano, June 28
6
7 Contents Introduction 9 Credit quality and credit risk 5. Default probability and loss given default Dierent approaches to credit risk modeling Credit migration Modeling dependence Portfolio credit risk Basic denitions and concepts Basel II insights Conditionally independent models Mixture models Threshold models A view on dependent credit migration models Contagion models Large portfolio losses Conditionally independent static models A static model with local interaction A conditionally Markov dynamic model Dynamic models with random environment Exogenous (static) random environment Some general results on large deviations The model for contagion Implementation of a Large Deviation Principle A law of large numbers for portfolio losses Simulation results
8 8 3.5 A central limit theorem Endogenous (dynamic) random environment The model in details Invariant measures and nonreversibility Studying the dynamics of the system Deterministic limit: Law of large numbers Equilibria of the limiting dynamics: Phase transition Fluctuations: A central limit theorem Convergence of generators approach A functional approach Applications to portfolio losses Simulation results Conclusions 5 A Technical proofs (Chapters 34) 7 A. Proof of Theorem A.2 Proof of Theorem A.3 Proof of Theorem A.4 Proof of Proposition A.5 Proof of Proposition A.6 The eigenvalues of the matrix A Bibliography 3
9 Introduction This Ph.D. thesis is part of a research project originated within the Probability group of the Mathematics department of the University of Padova in the year 25. This project involves the Professors Paolo Dai Pra and Wolfgang J. Runggaldier, a Ph.D. student of the University of Padova, Elena Sartori and Marco Tolotti of the Bocconi University and the Scuola ormale Superiore. The goal of this project is to integrate techniques and skills coming from dierent disciplines (Probability, Statistical Mechanics, Finance, Econometrics, Physics) in order to describe, analyze and quantify phenomena related to current issues in the Finance literature. Financial motivations The crucial issue that has motivated in particular this thesis is to describe a mathematical framework within which it is possible to explain the nancial phenomenon referred to clustering of defaults or credit crisis. By clustering of default we mean the situation in which many obligors experience nancial distress (default or downgrading in a rating system) in a short time period. What we mean by many defaults in a short time will be discussed later. It is clear that in order to speak of a credit crisis there must be an unexpected breakdown from the standard economic business cycle. In some sense we have in mind a sudden change in the equilibrium of the credit market. Financial distress, default, downgrading are all issues belonging to the eld of credit risk management. Managing the risk concerns the identication and the analysis of the randomness intrinsic in the nancial world and in particular the capacity to predict and quantify the losses triggered by changes in the variables describing the nancial market. More specically, when speaking of credit risk, one is dealing with the risks connected with the possible changes in the credit worthiness of the obligors. We shall concentrate, in particular, on the losses related to large portfolios, meaning to portfolios of many obligors with similar characteristics. The precise meaning of words such as large and similar shall be extensively discussed. When dealing with many obligors, the issue of modeling the dependence structure plays a major role. Our idea is that a credit crisis could be explained as the eect of a contagion process. A rm experiencing nancial distress may aect the credit quality of business 9
10 ITRODUCTIO partners (via direct contagion) as well as of rms in the same sector (due to an information eect). Therefore the mechanism of credit contagion is the crucial mechanism that we are going to develop in order to describe clustering of defaults. Reduced form models for direct contagion can be found among others in Jarrow and Yu (2) [47] for counterparty risk, Davis and Lo (2) [22] for infectious default, Kiyotaki and Moore (987) [48], where a model of credit chain obligations leading to default cascade is considered, Horst (26) [46] for domino eects, and Giesecke and Weber (25) [4] for a particle system approach. Concerning the banking sector, a microeconomic liquidity equilibrium is analyzed by Allen and Gale (2) []. Recent papers on information driven default models are e.g. Schönbucher (23) [59], Due et al. (26) [28], CollinDufresne et al. (23) [2]. An important point that we would like to stress is that we are aiming at describing the formation of a credit crisis starting from the inuences that the single obligors have among each others, in other words as a microeconomic phenomenon. The standard literature suggests that the aggregate nancial health of the system is fully described by some macroeconomic factors that capture the business cycle and then inuence the credit quality of the obligors. These factors are usually exogenously specied. One consequence of our micro" point of view is that in our modeling framework a global health indicator is endogenously computed and not a priori assigned. otice that the same philosophy is already standard practice in other disciplines: similar models and techniques as the ones developed in this thesis are applied for instance by Brock and Durlauf in (2) [8] for modeling social interactions or by Cont and Löwe (998) [5] in order to describe phenomena as herding behavior or peer pressure. The last remark is about a new trend in the literature concerning portfolio credit risk. In the last years portfolio credit derivatives such as default basket and Collateralized Debt Obligations (CDO's) have become very popular. In order to treat this kind of structured derivatives a dynamic study of the aggregate losses L(t) caused by the defaults in the underlying pool of assets becomes crucial. It has been documented that an eective study of the dynamics of the aggregate losses L(t) may not necessarily require a full understanding of the single name processes related to the underlying assets. For this reason a new approach has been recently proposed in the literature: the so called topdown approach (for more details see Cont and Minca (28) [6], Giesecke and Goldberg (27) [39] and Schönbucher (26) [6]). When considering Markov chain models similar in spirit to the ones proposed in this work, it is possible under certain hypothesis to fully explain the evolution of the system via aggregate sucient statistics. As argued also in Frey and Backhaus (27) [36], this is exactly the philosophy behind a topdown model. In particular we shall see in Chapter 3 (see Remark 3.5.8) that our approach may be considered as a useful tool also under this new perspective since it naturally exploits the problem of computing approximations of L(t) in a parsimonious way as a function of aggregate asymptotic variables that may account for the heterogeneity in the underlying portfolio. We have thus identied the nancial core of this discussion: quantifying the losses connected to the deterioration of credit quality, taking the contagion into account and eventually describing under which market conditions a credit crisis may take place.
11 Technical aspects From a more technical point of view there are dierent tasks that we have to deal with; we briey illustrate the most important ones: Having large portfolios in mind, we are going to describe asymptotic results for a system of innitely many rms and then provide nite volume approximations. It basically implies the development of suitable Laws of Large umbers (LL) and Central Limit Theorems (CLT). The implementation of LL and CLT for the study of large portfolios is not new in the risk management literature; for example it is implemented (in a rather basic setup, without considering contagion) in the Basel II accord. More recent generalizations to contagion models have been proposed among othersby Frey and Backhaus (26) [36] and by Giesecke and Weber (25) [4]. A dierent approach to the study of large portfolio losses may concern the extreme events (analyzing the tails of the loss distributions). In this case, it is quite common to rely on Large Deviation techniques. For a recent survey on these techniques, applied to Finance and risk management, see Pham (27) [56]. In Dembo et al. (26) [23] an application to large portfolio losses is proposed. The issue of contagion may be described relying on interacting particle systems borrowed from Statistical Mechanics. otice that also in the literature of quantitative risk management the terminology interacting intensities is used in order to describe reduced form models where interaction is taken into account. The use of particle systems is rather common in the Social Sciences literature, for instance when modeling social interactions and aggregate behaviors (see the paper by Cont and Bouchaud (2) [5] for a discussion on herding behavior in Finance). Particle and dynamical systems can be found also in the literature on nancial market modeling. It has been shown that some of these models have "thermodynamic limits" that exhibit similar features compared to the limiting distributions (in particular when looking at the tails) of market returns time series. For a discussion on nancial market modeling see the survey by Cont (999) [4] and the papers by Föllmer (994) [33] and Föllmer et al. (24) [34] that contain inspiring discussions on interacting agents. In order to derive a LL and a CLT for a particle system we rely on Large Deviation techniques. The idea is to consider M the space of probability measures on trajectories endowed with the Skorohod topology. It is quite easy to state a Large Deviation Principle (LDP) for a reference system where there is no interaction between the rms. Then the goal is to nd a suitable function F : M R + that relates (via Varadhan's lemma) the LDP for the reference system with our interactive model for contagion. This technique has been applied to spinip systems by Dai Pra and Den Hollander (996) [8]. Concerning the central limit theorem we shall develop two dierent approaches. The rst one (based on large deviations) relies on a theory by Bolthausen (986) [4]. The second one is based on a weak convergencetype approach based on the uniform convergence of the generators of the associated Markov chains (as developed in the book by Ethier and Kurtz (996) [32]).
12 2 ITRODUCTIO We are aiming at building a dynamic model. This has to be done in order to describe the time evolution of the variables describing the credit quality of the obligors, and hence (under particular conditions) the formation of a credit crisis. We shall propose a model where the credit crisis is connected with the existence of multiple equilibria for the dynamical system. The system may in fact spend some time near an unstable conguration and then suddenly decay to a stable one. We shall see that this eect is related to the phase transition, i.e., on the level of interaction in the model. It is nally worth to spend some words on the point of view that we have adopted in this research project (at least in its rst step). We have focused our attention on the modeling aspects, trying to build a model as simple as possible where the eects of contagion (as the credit crises) could have been observed. Moreover we have looked for a completely solvable model, where closed form solutions can be provided. Put dierently, we have given more relevance to the technical part of the problem and the modeling aspects. On the other hand we have not developed the validation part of the model, i.e., the calibration and the analysis of real data. evertheless many numerical simulations are provided in order to illustrate the shape of the loss distributions and the trajectories of the health indicators under dierent market conditions. Although this observation could be seen as a drawback of our work, we would like to argue that what we are aiming at are the qualitative aspects more than the quantitative ones (at least in the st step of the research). Indeed, the models we are going to propose are very basic and have to be considered as a starting point for the construction of more realistic models that may better t real data. Structure of the thesis and main results The thesis is divided into ve chapters and one appendix. Chapter and Chapter 2 are devoted to the introduction of the main concepts and the basic tools for managing credit risk. In writing this introductive part we had two goals in mind: rstly we aimed at letting this work be as much as possible self consistent. On the other hand we have tried to report and briey discuss the main approaches and models used in the literature. We have focused in particular on those models that can be considered as the starting point for our research. The guideline to these two chapters may be summarized in the three following fundamental questions: How can we model the credit quality of one obligor? How can we determine her default probability and the possible losses that the event of default may trigger? Given a set of dierent obligors, how can dependence be modeled and eventually joint default probabilities computed? How can we model credit contagion? Is it possible to build a model that explains clustering of defaults (credit crises)?
13 3 These questions shall be picked up again in more detail in the two introductive chapters and should nd appropriate answers in the further exposition. otice that they go from the very basic concepts up to the issues that have motivated our research. Hence these questions (and the two chapters themselves) are intended to build a bridge between the existing literature and our point of view. The succeeding two chapters contain the mathematical implementation and the discussion of our two original models, developed in order to tackle the problem of contagion in a credit risk perspective. In Chapter 3 we propose in particular a rst attempt to model an interactive system of defaultable counterparties where a local random environment enters into play. We shall see that a crucial object of this work is the so called empirical measure. Indeed, suppose that rms are acting in a market and their default indicators σ(t) = (σ i (t); i =,..., ), where σ i (t) {, }, evolve in time. We denote by σ[, T ] D[, T ] a trajectory on the interval [, T ] of such a rating indicator. D[, T ] denotes the space of càdlàg functions endowed with the Skorohood topology. The empirical measure ρ is dened as follows ρ (σ[, T ]) = δ (σi [,T ]). It is a random measure taking values in M (D[, T ]), the space of probabilities on D[, T ]. ρ weights the realizations of the dimensional process σ[, T ]. Put differently the empirical measure can be thought of the physical (historical) measure of the market. Most of the mathematical results of this thesis are concerned with the sequence of measures (ρ ). In this chapter we state in particular a large deviation principle for this sequence of measures (Theorem 3.3.3) and then we prove a suitable law of large numbers (Theorem 3.3.6), nding a unique Q such that ρ Q almost surely. Signicant and very useful for applications is also the characterization of Q provided in Proposition In Section 3.5 we nally state and prove a functional central limit theorem characterizing the uctuations of ρ around Q (Theorem 3.5.6). The proof of this theorem is inspired by the seminal work of E. Bolthausen (see [4]). In [4] a rather general framework is proposed in order to derive central limit theorems in Banach spaces. Since M is not a Banach space we are forced to construct an auxiliary space where a suitable large deviation principle is inherited and a central limit theorem can be proved accordingly. This procedure is summarized in Theorems and Although this rst model shows some interesting features and to some extentgeneralizes the present literature on dynamic meaneld models for large portfolio losses, it turns out to be not yet comprehensive enough in order to explain clustering and credit crises.
14 4 ITRODUCTIO Chapter 4 is then devoted to the analysis of a new framework that makes possible the explicit identication of the desired clustering eect. To this aim we introduce a fundamental indicator of robustness ω i {, } that is coupled with σ i dened before. The 2 state variables (σ, ω) evolve in time and their dynamics turn out to be non trivial at all. In particular our model is nonreversible. As compared to similar but reversible stochastic interacting systems, more careful arguments have to be used in order to prove the large deviation principle, which represents the basic tool in our approach. Similarly to Chapter 3, the rst main result is a Law of Large umbers (Theorem 4.3.2) based on a Large Deviation Principle (Proposition 4.3.4). In Theorem 4.3. we shall see that dierent asymptotic congurations can be found, depending on the values of the parameters. This phenomenon (called phase transition) has implications for the description of a credit crisis as we shall explain in more details in Chapter 5. The last section of this chapter is devoted to the study of the uctuations of the empirical measure around its limit. Two dierent approaches are described, the former is based on uniform convergence of generators (Theorem 4.4.). The latter (Theorem 4.4.5) mimics the functional approach already introduced in the previous chapter. One remark is needed at this point. Part of these rather technical proofs have been pursued in collaboration with Paolo Dai Pra and Elena Sartori. We shall refer to the Ph.D. thesis of Elena where some explicit computations have not been reported in this dissertation. To make the exposition less heavy we have postponed to Appendix A the most technical proofs of Chapters 3 and 4. The nancial applications are discussed in Chapter 5. The main result of this chapter is Theorem It is concerned with the computation of risk measures for managing the risk involved in large portfolios. Various examples are provided, some of them have been suggested by the existing literature on the subject. The second issue that we are going to analyze in this chapter is related to the formation of a credit crisis. To this aim the equilibria of the limiting dynamics and their stability are studied and the phase transition is fully characterized. At the end of the chapter, dierent graphs and numerical implementations are presented in order to support the nancial interpretation of our model. Finally we conclude this thesis with a brief summary of the main results and mentioning some open problems and possible lines of future research in this area.
15 Chapter Credit quality and credit risk The goal of the rst two chapters of this thesis is to equip the reader with the basic notations and concepts necessary to enter the world of credit risk. First of all we would like to specify what we intend for "risk" in the context of credit and propose a mathematical framework in which a punctual quantitative analysis can be pursued. A rather concise denition of risk in nance could be as follows: we speak about risk when we consider the possibility of having unexpected changes in the variables that describe the nancial model. We thus have to dene a suitable probability space {Ω, F, P } that summarizes the states of the world, the interesting events and a possible probability measure on them. We may possibly add a ltration (F t ) t that describes the ow of information when considering dynamic models. Finally we dene random variables (eventually processes) X : Ω R, representing the nancial variables that we are interested in. In the context of credit risk, X should basically describe the credit quality of a given obligor, where an obligor is somebody who has to pay back a debt to somebody else in the future. For credit quality we mean the ability of being able to pay back obligations. How can we model the credit quality of one obligor? How can we determine her default probability and the possible losses that the event of default may trigger? Suppose that this obligor is a rm; in this case X could represents the value of the rm at a given time. Thus we could be interested in estimating the probability that X falls below a certain value (also named a threshold level) and this could be a good choice for a credit quality indicator. In an even naiver world, X could simply be an indicator of default (bankruptcy): if X = the rm is able to pay back its obligation, if X = it is not. All these issues will be developed and specied in the following chapters. In particular many dierent models with dierent specications for X will be provided as well as dierent specications for {Ω, F, P } and (F t ) t. We would like to stress the fact that the majority of the concepts we are going to state in the rst two chapters are not new and can be found in dierent books dealing with credit risk. Our purpose is to give a glance on the main aspects, focusing the attention on the basic building blocks necessary to assess and quantify the riskiness of a portfolio of obligors and to price credit derivatives. We shall describe the existing techniques used to compute these building blocks and the relevance of particular 5
16 6 CHAPTER. CREDIT QUALITY AD CREDIT RISK assumptions often used to provide explicit formulae. In doing this we shall build the bridge between existing literature and our modeling ideas, leading the reader to the new framework that we have introduced in order to solve some open problems that are still debated in the present literature.. Default probability and loss given default In this section we briey recall the very basic tools used to deal with credit risk. Our aim is to focus the attention on the so called building blocks, i.e., the basic objects on which the risk measurement and the pricing techniques of large part of more complicated securities rely on. These objects are the so called default probability and the loss given default. In a credit risk environment the basic concept is default. We do not enter into the discussion of the bankruptcy procedures. We simply say that a default time τ has to be dened. In the next section we shall discuss on how τ is characterized within dierent models. Our concern now is simply that τ is a random time and in particular a stopping time with respect to its own ltration (H t ) t. Default probability (PD) We dene the default probability between t and T for the obligor as p(t, T ) = P (τ T τ > t). (.) It is worth to dene also the conditional version for the default probabilities meaning the probability at time t of having a default between future dates T and T 2 knowing survival up to T : p(t, T, T 2 ) = p(t, T 2) p(t, T ), t T T 2. These quantities are then used in order to determine an important object often referred to the hazard rate. We dene the hazard rate between t and T as h(t, T ) = lim t p(t, T, T + t) t whenever this limit exists. Remark.. It should be stressed that in more general models (for example if the interest rate r is stochastic or if other random variables inuence the market prices) one has to be careful with the denition of hazard rates. In particular it is very important to study measurability conditions (namely which ltration or information structure are available in the model) and how the total information is related to the ltration generated by the default process (see chapters 5,6 in [3] for a punctual discussion). In particular it can be shown (see for instance Section 6.2 in [3]) that for a random time τ one may consider two dierent hazard processes (both well dened and mathematically meaningful) and they may not coincide if some regularity hypotheses are not made on the model. How F and H are related is a crucial point. An extensive discussion is made in [3]. We do not enter into details at this point.
17 .2. DIFFERET APPROACHES TO CREDIT RISK MODELIG 7 Formally we can also dene a local hazard rate: γ(t) := h(t, t) = lim p(τ t + t τ > t). t t Loss given default (LGD) Suppose that a default happens. Suppose moreover that a bank holds a position issued by the defaulting obligor. At the time of default τ < T part of the investment of the bank is lost due to the impossibility of the obligor of paying back obligations. The part that the bank (or more general an investor) cannot recover is called loss given default. This quantity is often modeled as a random variable l = δe where δ represents a random proportion of the exposure that is lost and e the actual exposure at default. A rather general discussion on pricing issues of defaultable claims is also proposed in [3]. There, it is shown that (PD) and (LGD) are the building blocks, necessary to price many types of derivatives such as defaultable xedcoupon bonds, credit default swaps (and many variants of them), asset swap packages..2 Dierent approaches to credit risk modeling Up to now we have considered a given probability space {Ω, F, (F t ) t, P } endowed with a market ltration. On this ltered probability space we have dened a default time τ without taking care of the economic process leading to this denition and without specifying any distributional assumption on it (except for some assumptions on the corresponding ltration). We now have to implement models that may help in computing the building blocks seen in the previous section. In the literature two typical and rather dierent point of views are developed. The rst class of models, the so called structural models (or rm value models), relies on the precise denition of some economical variables (such as the asset value process for a rm). The evolution of these variables determine the credit quality of the obligor itself. The second class of models, the reduced form models (or intensity based models), is based on a more statistical point of view. The idea here is that having dened a suitable family of processes (without direct economical interpretation) one tries to calibrate the parameters in order to t historical time series or other data. These two approaches lead of course to dierent modeling frameworks. Both have important advantages: the former gives more direct intuition on what is going on economically; the latter is usually easier to implement and allows for more freedom in modeling parameters and in tting data 2. 2 We would like to stress the fact that the dilemma" on what the best philosophy" should be, is far from being solved. Moreover, notice that it involves the fundamental debate whether when using random variables one is or is not allowed to forget" about the underlying probability space (the underlying experiment") taking into account only distributional consequences (the numerical results of the experiment).
18 8 CHAPTER. CREDIT QUALITY AD CREDIT RISK A. Structural models The basic idea behind structural models is to consider the evolution in time of an underlying process X that is related to some fundamental indicators of the rm. Then, usually, the default happens when this process hits a predetermined (possibly stochastic) barrier D. So that τ is dened as a rst passage stopping time, indeed τ = inf t {X t D}. (.2) In the progenitor of these models, due to Merton (974) [53], the underlying process was the rm asset value process V t. In particular it is assumed that V t = S t + B t where S t is the value at time t of the equity and B t is the value at time t of a zero coupon bond with face value B and maturity T. In this basic context where the payments to the bondholders are due at a xed time T, the event of default is even easier to describe, than in Equation (.2). We have in fact that default happens if and only if V T < D so that τ = T. Moreover the recovery at default is simply B T = min(v T, B). Suppose that V t evolves according to the dierential equation dv t = µ V V t dt + σ V V t dw t where V = V >, µ V R and σ V > are constants and W is a standard Brownian motion. Then one can compute the default probability ( ln(b/v ) (µv σv 2 p(, T ) = P (V T < B) = /2) ) σ V T where ( ) stands for the standard normal distribution function. We do not want to enter into more details on this topic, we simply mention the fact that this modeling idea is still used in practice. More sophisticated models have been proposed, we shall see some extensions in the next sections (the KMV model, credit migration models, a multivariate extension of Merton model,...). What remains a milestone in all of them is the precise reference to an underlying explanatory (fundamental) process and the fact that the default time is an hitting time 3. B. Reduced form models We have seen what a model for credit risk should be able to provide: probabilities of default and losses given default. In the structural approach, we have seen a basic methodology to compute them. One dierent approach is to consider models where the family of the distributions for the probabilities of default is a priori given and the model parameters are then 3 The fact that τ is an hitting time for a continuous path process makes the time τ be fully predictable. In particular this implies that the local probability of default h(t) is always zero. We shall see that this is not the case in the so called intensity based models (and this reects better real data). This is probably the main (mathematical) dierence between the two approaches and the reason that makes intensity based models so popular. Many authors have tried to relax this hypothesis in order to make τ totally inaccessible. We refer to [58] for a discussion on the accessibility of a stopping time and to [22], [], [9], [43] and [44] for dierent bridge" models between structural and reduced form models.
19 .3. CREDIT MIGRATIO 9 computed in order to t market data. In particular it is useful to consider models where a local probability of default is well dened. As in the previous sections we assume that a probability space {Ω, F, P } is given. We consider the nonnegative random variable τ : Ω R +, with P {τ = } = and P {τ > t} > for all t, that describes the default time. For simplicity we suppose that the information available to the market is the natural ltration generated by the default time τ. Indeed, we consider the σelds H t := σ({τ u} : u t) and the corresponding ltration H = (H t ) t. In many reduced form models the default time τ is distributed according to an exponential law, so that the event of default is related to the rst jump of a Poisson process ((t)) t such that τ = inf{t : (t) = }. After the rst jump we stop the Poisson process so that for t > τ we have (t) = (τ) =. Summarizing we nally have (t) = (t τ) = I {τ t}. otice that H t = σ((u) : u t). We denote by F (t) = P {τ t} for all t R + the cumulative distribution function of the default time. Assumption.2. F (t) is absolutely continuous with respect to the Lebesgue measure, that is, it admits a density function f(t) : R + R + such that F (t) = t f(u)du, then F (t) = e t γ(u)du, where γ(t) = f(t)( F (t)). The function γ is called intensity function. It can be shown that in this case γ(t) := lim P (τ t + t τ > t). t t Put dierently, it describes the local probability of default. In the class of models where the intensity function is well dened, it is natural to assume γ as the primitive object to be characterized. These models are usually called intensity based models..3 Credit migration In the previous sections we have described the basic tools when dealing with default risk. In particular we have dened a stopping time τ as the time of default. We want now to consider more general frameworks in which the default event is no longer the only interesting event (and the unique risk to be taken into account). Instead of looking only at the default event, we shall describe the credit quality of an obligor considering a whole set of rating classes where the default is only the last (and worst) class. We are thinking of the so called credit rating models. These models are implemented by rating agencies (Moody's, Standard and Poor's, Fitch) but also by the internal rating systems of nancial insitutions 4. The basic idea behind these models is to assign each obligor to one class that is characterized by an estimated probability of default (i.e. of falling down to the default state). This means that all the available information about the probability of default of each obligor is given by its rating class. 4 In the new Basel II accord the single institutions are encouraged to implement internal systems to asses credit worthiness of obligors, see [2].
20 2 CHAPTER. CREDIT QUALITY AD CREDIT RISK For example in Moody's model the rating classes are labelled from the best to the worst as AAA, AA, A, BBB,..., CCC, D, where D indicates the default state. Thus we have to estimate a Kdimensional vector of probabilities of default where K is the number of rating classes (apart form default). otice that all the migration probabilities have to be determined, in the sense that we are interested in the probability that a rm in class AA falls within a year into class B, and this has to be done for each pair of classes. This means that we have to determine a transition matrix in which the entries are exactly the probabilities of migration among the K + classes. To formalize this issue we give the following denitions: Denition.3. We label the K + classes as K, K,...,, where k = indicates the default state; We name K the set of all the rating classes and K = K /{} the set of all the classes except for default; Dene the rating process R {t } as the process R(ω, t) : Ω [, ] K where {Ω, F, F t, P } is a probability space where F t is the market ltration; We dene Q(t, T ) R K+ R K+ as the transition probability matrix for the time interval [t, T ], so that (Q(t, T )) ij = P (R(T ) = j F t R(t) = i). otice that (Q(t, T )) ij, i, j K ;, j K and Q(t, t) = I where I is the identity matrix. K j= Q(t, T ) i,j =, i K ; Q(t, T ),j = The problem of credit migration is now to specify a model in order to describe the evolution of the transition matrix Q(t, T ) and eventually to calibrate it to the data. In the standard literature as well as in the most famous models used in practice, the rating transition process R is assumed to evolve as a time homogeneous Markov chain. Basically this means that (Q(t, T )) ij = P (R(T ) = j R(t) = i); Q(t, T ) = Q(T t) t T. where Q is a transition probability that depends only on the length of the time interval between t and T. The Markov hypothesis simplies the computation of the migration probabilities. It can be proved (see [58] for more details) that in this case Q(t, T ) = exp{(t t)λ} where the exponential is dened formally as the limit of the series expansion exp{(t t)λ} = ((T t)λ) n n= n!
Bilateral Exposures and Systemic Solvency Risk
Bilateral Exposures and Systemic Solvency Risk C., GOURIEROUX (1), J.C., HEAM (2), and A., MONFORT (3) (1) CREST, and University of Toronto (2) CREST, and Autorité de Contrôle Prudentiel et de Résolution
More informationCredit Risk Models: An Overview
Credit Risk Models: An Overview Paul Embrechts, Rüdiger Frey, Alexander McNeil ETH Zürich c 2003 (Embrechts, Frey, McNeil) A. Multivariate Models for Portfolio Credit Risk 1. Modelling Dependent Defaults:
More informationPortfolio Distribution Modelling and Computation. Harry Zheng Department of Mathematics Imperial College h.zheng@imperial.ac.uk
Portfolio Distribution Modelling and Computation Harry Zheng Department of Mathematics Imperial College h.zheng@imperial.ac.uk Workshop on Fast Financial Algorithms Tanaka Business School Imperial College
More informationProbability Models of Credit Risk
Probability Models of Credit Risk In discussing financial risk, it is useful to distinguish between market risk and credit risk. Market risk refers to the possibility of losses due to changes in the prices
More informationStatistics for Retail Finance. Chapter 8: Regulation and Capital Requirements
Statistics for Retail Finance 1 Overview > We now consider regulatory requirements for managing risk on a portfolio of consumer loans. Regulators have two key duties: 1. Protect consumers in the financial
More informationEconomic Catastrophe Bonds: Inecient Market or Inadequate Model?
Economic Catastrophe Bonds: Inecient Market or Inadequate Model? Haitao Li a and Feng Zhao b ABSTRACT In an inuential paper, Coval, Jurek and Staord (2009, CJS hereafter) argue that senior CDX tranches
More informationMiscellaneous. Simone Freschi freschi.simone@gmail.com Tommaso Gabriellini tommaso.gabriellini@mpscs.it. Università di Siena
Miscellaneous Simone Freschi freschi.simone@gmail.com Tommaso Gabriellini tommaso.gabriellini@mpscs.it Head of Global MPS Capital Services SpA  MPS Group Università di Siena ini A.A. 20142015 1 / 1 A
More informationHedging of Life Insurance Liabilities
Hedging of Life Insurance Liabilities Thorsten Rheinländer, with Francesca Biagini and Irene Schreiber Vienna University of Technology and LMU Munich September 6, 2015 horsten Rheinländer, with Francesca
More informationFORWARDS AND EUROPEAN OPTIONS ON CDO TRANCHES. John Hull and Alan White. First Draft: December, 2006 This Draft: March 2007
FORWARDS AND EUROPEAN OPTIONS ON CDO TRANCHES John Hull and Alan White First Draft: December, 006 This Draft: March 007 Joseph L. Rotman School of Management University of Toronto 105 St George Street
More informationCAPM, Arbitrage, and Linear Factor Models
CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose meanvariance e cient portfolios. By equating these investors
More informationArticle from: Risk Management. June 2009 Issue 16
Article from: Risk Management June 2009 Issue 16 CHAIRSPERSON S Risk quantification CORNER Structural Credit Risk Modeling: Merton and Beyond By Yu Wang The past two years have seen global financial markets
More informationLECTURE 10.1 Default risk in Merton s model
LECTURE 10.1 Default risk in Merton s model Adriana Breccia March 12, 2012 1 1 MERTON S MODEL 1.1 Introduction Credit risk is the risk of suffering a financial loss due to the decline in the creditworthiness
More informationMonte Carlo Simulation
1 Monte Carlo Simulation Stefan Weber Leibniz Universität Hannover email: sweber@stochastik.unihannover.de web: www.stochastik.unihannover.de/ sweber Monte Carlo Simulation 2 Quantifying and Hedging
More informationExpected default frequency
KM Model Expected default frequency Expected default frequency (EDF) is a forwardlooking measure of actual probability of default. EDF is firm specific. KM model is based on the structural approach to
More informationThe Time Value of Money
The Time Value of Money This handout is an overview of the basic tools and concepts needed for this corporate nance course. Proofs and explanations are given in order to facilitate your understanding and
More informationIMPLEMENTATION NOTE. Validating Risk Rating Systems at IRB Institutions
IMPLEMENTATION NOTE Subject: Category: Capital No: A1 Date: January 2006 I. Introduction The term rating system comprises all of the methods, processes, controls, data collection and IT systems that support
More informationINDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)
INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulationbased method for estimating the parameters of economic models. Its
More informationLOGNORMAL MODEL FOR STOCK PRICES
LOGNORMAL MODEL FOR STOCK PRICES MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD 1. INTRODUCTION What follows is a simple but important model that will be the basis for a later study of stock prices as
More informationThe Basics of Interest Theory
Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest
More informationSchonbucher Chapter 9: Firm Value and Share PricedBased Models Updated 07302007
Schonbucher Chapter 9: Firm alue and Share PricedBased Models Updated 07302007 (References sited are listed in the book s bibliography, except Miller 1988) For Intensity and spreadbased models of default
More informationNon Linear Dependence Structures: a Copula Opinion Approach in Portfolio Optimization
Non Linear Dependence Structures: a Copula Opinion Approach in Portfolio Optimization Jean Damien Villiers ESSEC Business School Master of Sciences in Management Grande Ecole September 2013 1 Non Linear
More informationBlackScholesMerton approach merits and shortcomings
BlackScholesMerton approach merits and shortcomings Emilia Matei 1005056 EC372 Term Paper. Topic 3 1. Introduction The BlackScholes and Merton method of modelling derivatives prices was first introduced
More informationLife Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans
Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans Challenges for defined contribution plans While Eastern Europe is a prominent example of the importance of defined
More information1 Example of Time Series Analysis by SSA 1
1 Example of Time Series Analysis by SSA 1 Let us illustrate the 'Caterpillar'SSA technique [1] by the example of time series analysis. Consider the time series FORT (monthly volumes of fortied wine sales
More informationApplication of Quantitative Credit Risk Models in Fixed Income Portfolio Management
Application of Quantitative Credit Risk Models in Fixed Income Portfolio Management Ron D Vari, Ph.D., Kishore Yalamanchili, Ph.D., and David Bai, Ph.D. State Street Research and Management September 263,
More information4. Only one asset that can be used for production, and is available in xed supply in the aggregate (call it land).
Chapter 3 Credit and Business Cycles Here I present a model of the interaction between credit and business cycles. In representative agent models, remember, no lending takes place! The literature on the
More information6. Budget Deficits and Fiscal Policy
Prof. Dr. Thomas Steger Advanced Macroeconomics II Lecture SS 2012 6. Budget Deficits and Fiscal Policy Introduction Ricardian equivalence Distorting taxes Debt crises Introduction (1) Ricardian equivalence
More informationMULTIPLE DEFAULTS AND MERTON'S MODEL L. CATHCART, L. ELJAHEL
ISSN 17446783 MULTIPLE DEFAULTS AND MERTON'S MODEL L. CATHCART, L. ELJAHEL Tanaka Business School Discussion Papers: TBS/DP04/12 London: Tanaka Business School, 2004 Multiple Defaults and Merton s Model
More informationMaster of Mathematical Finance: Course Descriptions
Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support
More informationRegulatory and Economic Capital
Regulatory and Economic Capital Measurement and Management Swati Agiwal November 18, 2011 What is Economic Capital? Capital available to the bank to absorb losses to stay solvent Probability Unexpected
More informationChap 3 CAPM, Arbitrage, and Linear Factor Models
Chap 3 CAPM, Arbitrage, and Linear Factor Models 1 Asset Pricing Model a logical extension of portfolio selection theory is to consider the equilibrium asset pricing consequences of investors individually
More informationMarshallOlkin distributions and portfolio credit risk
MarshallOlkin distributions and portfolio credit risk Moderne Finanzmathematik und ihre Anwendungen für Banken und Versicherungen, Fraunhofer ITWM, Kaiserslautern, in Kooperation mit der TU München und
More informationA Simple Model of Price Dispersion *
Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 112 http://www.dallasfed.org/assets/documents/institute/wpapers/2012/0112.pdf A Simple Model of Price Dispersion
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More informationHydrodynamic Limits of Randomized Load Balancing Networks
Hydrodynamic Limits of Randomized Load Balancing Networks Kavita Ramanan and Mohammadreza Aghajani Brown University Stochastic Networks and Stochastic Geometry a conference in honour of François Baccelli
More information10. FixedIncome Securities. Basic Concepts
0. FixedIncome Securities Fixedincome securities (FIS) are bonds that have no default risk and their payments are fully determined in advance. Sometimes corporate bonds that do not necessarily have certain
More informationUlrich A. Muller UAM.19940131. June 28, 1995
Hedging Currency Risks { Dynamic Hedging Strategies Based on O & A Trading Models Ulrich A. Muller UAM.19940131 June 28, 1995 A consulting document by the O&A Research Group This document is the property
More informationFactor models and the credit risk of a loan portfolio
MPRA Munich Personal RePEc Archive Factor models and the credit risk of a loan portfolio Edgardo Palombini October 2009 Online at http://mpra.ub.unimuenchen.de/20107/ MPRA Paper No. 20107, posted 18.
More informationThesis work and research project
Thesis work and research project Hélia Pouyllau, INRIA of Rennes, Campus Beaulieu 35042 Rennes, helia.pouyllau@irisa.fr July 16, 2007 1 Thesis work on Distributed algorithms for endtoend QoS contract
More informationRecoveries on Defaulted Debt
Recoveries on Defaulted Debt Michael B. Gordy Federal Reserve Board May 2008 The opinions expressed here are my own, and do not reflect the views of the Board of Governors or its
More informationMarketing Mix Modelling and Big Data P. M Cain
1) Introduction Marketing Mix Modelling and Big Data P. M Cain Big data is generally defined in terms of the volume and variety of structured and unstructured information. Whereas structured data is stored
More informationIntroduction to time series analysis
Introduction to time series analysis Margherita Gerolimetto November 3, 2010 1 What is a time series? A time series is a collection of observations ordered following a parameter that for us is time. Examples
More informationTopic 1: Matrices and Systems of Linear Equations.
Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method
More informationVasicek Single Factor Model
Alexandra Kochendörfer 7. Februar 2011 1 / 33 Problem Setting Consider portfolio with N different credits of equal size 1. Each obligor has an individual default probability. In case of default of the
More informationIEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem
IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have
More informationCREDIT RISK MANAGEMENT
GLOBAL ASSOCIATION OF RISK PROFESSIONALS The GARP Risk Series CREDIT RISK MANAGEMENT Chapter 1 Credit Risk Assessment Chapter Focus Distinguishing credit risk from market risk Credit policy and credit
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationGeometric Brownian motion makes sense for the call price because call prices cannot be negative. Now Using Ito's lemma we can nd an expression for dc,
12 Option Pricing We are now going to apply our continuoustime methods to the pricing of nonstandard securities. In particular, we will consider the class of derivative securities known as options in
More informationVilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis
Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................
More informationDistressed Debt Prices and Recovery Rate Estimation
Distressed Debt Prices and Recovery Rate Estimation Robert Jarrow Joint Work with Xin Guo and Haizhi Lin May 2008 Introduction Recent market events highlight the importance of understanding credit risk.
More informationDynamic Factor Copula Model
Dynamic Factor Copula Model Ken Jackson Alex Kreinin Wanhe Zhang July 6, 2009 Abstract The Gaussian factor copula model is the market standard model for multiname credit derivatives. Its main drawback
More informationDiscussion Paper On the validation and review of Credit Rating Agencies methodologies
Discussion Paper On the validation and review of Credit Rating Agencies methodologies 17 November 2015 ESMA/2015/1735 Responding to this paper The European Securities and Markets Authority (ESMA) invites
More informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More information2. Default correlation. Correlation of defaults of a pair of risky assets
2. Default correlation Correlation of defaults of a pair of risky assets Consider two obligors A and B and a fixed time horizon T. p A = probability of default of A before T p B = probability of default
More informationDiusion processes. Olivier Scaillet. University of Geneva and Swiss Finance Institute
Diusion processes Olivier Scaillet University of Geneva and Swiss Finance Institute Outline 1 Brownian motion 2 Itô integral 3 Diusion processes 4 BlackScholes 5 Equity linked life insurance 6 Merton
More informationSpatial Statistics Chapter 3 Basics of areal data and areal data modeling
Spatial Statistics Chapter 3 Basics of areal data and areal data modeling Recall areal data also known as lattice data are data Y (s), s D where D is a discrete index set. This usually corresponds to data
More informationCPC/CPA Hybrid Bidding in a Second Price Auction
CPC/CPA Hybrid Bidding in a Second Price Auction Benjamin Edelman Hoan Soo Lee Working Paper 09074 Copyright 2008 by Benjamin Edelman and Hoan Soo Lee Working papers are in draft form. This working paper
More informationQuantitative Operational Risk Management
Quantitative Operational Risk Management Kaj Nyström and Jimmy Skoglund Swedbank, Group Financial Risk Control S105 34 Stockholm, Sweden September 3, 2002 Abstract The New Basel Capital Accord presents
More informationRisk management of CPPI funds in switching regime markets.
Risk management of CPPI funds in switching regime markets. Donatien Hainaut October, 1 NSACRST. 945 Malako Cedex, France. mail: donatien.hainaut@ensae.fr Abstract The constant proportion portfolio insurance
More informationStochastic Analysis of LongTerm MultipleDecrement Contracts
Stochastic Analysis of LongTerm MultipleDecrement Contracts Matthew Clark, FSA, MAAA, and Chad Runchey, FSA, MAAA Ernst & Young LLP Published in the July 2008 issue of the Actuarial Practice Forum Copyright
More informationStatistics in Retail Finance. Chapter 6: Behavioural models
Statistics in Retail Finance 1 Overview > So far we have focussed mainly on application scorecards. In this chapter we shall look at behavioural models. We shall cover the following topics: Behavioural
More informationSTRUCTURED FINANCE RATING CRITERIA 2015
STRUCTURED FINANCE RATING CRITERIA 2015 1. Introduction 3 1.1 Credit quality of the collateral 3 1.2 The structure designed 3 1.3 Qualitative risks on the Securitization Fund 4 1.4 Sensitivity 4 1.5 Definition
More informationSupplement to Call Centers with Delay Information: Models and Insights
Supplement to Call Centers with Delay Information: Models and Insights Oualid Jouini 1 Zeynep Akşin 2 Yves Dallery 1 1 Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290
More informationBias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes
Bias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes Yong Bao a, Aman Ullah b, Yun Wang c, and Jun Yu d a Purdue University, IN, USA b University of California, Riverside, CA, USA
More informationMultidimensional BlackScholes options
MPRA Munich Personal RePEc Archive Multidimensional BlackScholes options Francesco Paolo Esposito 10 December 2010 Online at https://mpra.ub.unimuenchen.de/42821/ MPRA Paper No. 42821, posted 24 November
More informationAnalysis of a Production/Inventory System with Multiple Retailers
Analysis of a Production/Inventory System with Multiple Retailers Ann M. Noblesse 1, Robert N. Boute 1,2, Marc R. Lambrecht 1, Benny Van Houdt 3 1 Research Center for Operations Management, University
More informationThe Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models
780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond marketmaker would deltahedge, we first need to specify how bonds behave. Suppose we try to model a zerocoupon
More informationBlack Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869. Words: 3441
Black Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869 Words: 3441 1 1. Introduction In this paper I present Black, Scholes (1973) and Merton (1973) (BSM) general
More informationAsymmetry and the Cost of Capital
Asymmetry and the Cost of Capital Javier García Sánchez, IAE Business School Lorenzo Preve, IAE Business School Virginia Sarria Allende, IAE Business School Abstract The expected cost of capital is a crucial
More informationFINANCIAL ECONOMICS OPTION PRICING
OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.
More informationPortfolio Management for Banks
Enterprise Risk Solutions Portfolio Management for Banks RiskFrontier, our industryleading economic capital and credit portfolio risk management solution, along with our expert Portfolio Advisory Services
More informationExtending Factor Models of Equity Risk to Credit Risk and Default Correlation. Dan dibartolomeo Northfield Information Services September 2010
Extending Factor Models of Equity Risk to Credit Risk and Default Correlation Dan dibartolomeo Northfield Information Services September 2010 Goals for this Presentation Illustrate how equity factor risk
More informationThe Exponential Distribution
21 The Exponential Distribution From DiscreteTime to ContinuousTime: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding
More informationCONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE
1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,
More informationNonBank Deposit Taker (NBDT) Capital Policy Paper
NonBank Deposit Taker (NBDT) Capital Policy Paper Subject: The risk weighting structure of the NBDT capital adequacy regime Author: Ian Harrison Date: 3 November 2009 Introduction 1. This paper sets out,
More informationHedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies
Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative
More informationNonparametric adaptive age replacement with a onecycle criterion
Nonparametric adaptive age replacement with a onecycle criterion P. CoolenSchrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK email: Pauline.Schrijner@durham.ac.uk
More informationTHE CENTRAL LIMIT THEOREM TORONTO
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................
More informationInterlinkages between Payment and Securities. Settlement Systems
Interlinkages between Payment and Securities Settlement Systems David C. Mills, Jr. y Federal Reserve Board Samia Y. Husain Washington University in Saint Louis September 4, 2009 Abstract Payments systems
More informationSensitivity analysis of utility based prices and risktolerance wealth processes
Sensitivity analysis of utility based prices and risktolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationFinancial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan
Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction
More informationReview of Basic Options Concepts and Terminology
Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some
More informationDebt Instruments Set 3
Debt Instruments Set 3 Backus/February 9, 1998 Quantifying Interest Rate Risk 0. Overview Examples Price and Yield Duration Risk Management Convexity ValueatRisk Active Investment Strategies Debt Instruments
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationThe Trip Scheduling Problem
The Trip Scheduling Problem Claudia Archetti Department of Quantitative Methods, University of Brescia Contrada Santa Chiara 50, 25122 Brescia, Italy Martin Savelsbergh School of Industrial and Systems
More informationSEO Risk Dynamics. MurrayCarlson,AdlaiFisher,andRonGiammarino TheUniversityofBritishColumbia. December21,2009
SEO Risk Dynamics MurrayCarlson,AdlaiFisher,andRonGiammarino TheUniversityofBritishColumbia December21,2009 SauderSchoolofBusiness,UniversityofBritishColumbia,2053MainMall,Vancouver,BC,V6T1Z2.Wethank fortheirhelpfulcommentsfelipeaguerrevere,leoncebargeron,sugatobhattacharya,michaelbrandt,alonbrav,
More informationLecture Notes: Basic Concepts in Option Pricing  The Black and Scholes Model
Brunel University Msc., EC5504, Financial Engineering Prof Menelaos Karanasos Lecture Notes: Basic Concepts in Option Pricing  The Black and Scholes Model Recall that the price of an option is equal to
More informationarxiv:1412.1183v1 [qfin.rm] 3 Dec 2014
Regulatory Capital Modelling for Credit Risk arxiv:1412.1183v1 [qfin.rm] 3 Dec 2014 Marek Rutkowski a, Silvio Tarca a, a School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia.
More informationThe Western Hemisphere Credit & Loan Reporting Initiative (WHCRI)
The Western Hemisphere Credit & Loan Reporting Initiative (WHCRI) Public Credit Registries as a Tool for Bank Regulation and Supervision Matías Gutierrez Girault & Jane Hwang III Evaluation Workshop Mexico
More informationWhen to Refinance Mortgage Loans in a Stochastic Interest Rate Environment
When to Refinance Mortgage Loans in a Stochastic Interest Rate Environment Siwei Gan, Jin Zheng, Xiaoxia Feng, and Dejun Xie Abstract Refinancing refers to the replacement of an existing debt obligation
More informationThe Real Business Cycle Model
The Real Business Cycle Model Ester Faia Goethe University Frankfurt Nov 2015 Ester Faia (Goethe University Frankfurt) RBC Nov 2015 1 / 27 Introduction The RBC model explains the comovements in the uctuations
More informationThe VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.
Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium
More informationThe Empirical Approach to Interest Rate and Credit Risk in a Fixed Income Portfolio
www.empirical.net Seattle Portland Eugene Tacoma Anchorage March 27, 2013 The Empirical Approach to Interest Rate and Credit Risk in a Fixed Income Portfolio By Erik Lehr In recent weeks, market news about
More informationThe Binomial Option Pricing Model André Farber
1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a nondividend paying stock whose price is initially S 0. Divide time into small
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationModeling and simulation of a double auction artificial financial market
Modeling and simulation of a double auction artificial financial market Marco Raberto a,1 and Silvano Cincotti a a DIBE, Universit di Genova, Via Opera Pia 11a, 16145 Genova, Italy Abstract We present
More informationAn analysis of price impact function in orderdriven markets
Available online at www.sciencedirect.com Physica A 324 (2003) 146 151 www.elsevier.com/locate/physa An analysis of price impact function in orderdriven markets G. Iori a;, M.G. Daniels b, J.D. Farmer
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationSession IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics
Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock
More information1) What kind of risk on settlements is covered by 'Herstatt Risk' for which BCBS was formed?
1) What kind of risk on settlements is covered by 'Herstatt Risk' for which BCBS was formed? a) Exchange rate risk b) Time difference risk c) Interest rate risk d) None 2) Which of the following is not
More information